Dual integral equations involving Fourier transforms with increasing symbols (Q991556)

The following dual equation is considered \[ F^{-1}[|\xi|^p A(\xi)\hat{u}(\xi)](x)=f(x),\quad x\in(a,b), \] \[ F^{-1}[|\hat{u}(\xi)|](x)=0, \quad x\in \mathbb{R} \setminus (a,b). \] Here \(F\) and \(F^{-1}\) are the classical Fourier transforms, \(\hat{u}\) is the unknown function, \(p\) is an arbitrary non-negative integer. A method for investigating and solving this dual integral equation involving Fourier transform with increasing symbols is proposed. The above dual equation is then reduced to equivalent Fredholm equation of the second kind depending on whether \(p\) is an odd or even number.